Proof that the adjoint representation is an endomorphism

[atat1tata]

Homework Statement

My textbooks takes for granted that, given a Lie group $g$ and its algebra $\mathfrak{g}$, we have that $AXA^{-1} \in \mathfrak{g}$.

Homework Equations

For $Y$ to be in $\mathfrak{g}$ means that $e^{tY} \in G$ for each $t \in \mathbf{R}$

The Attempt at a Solution

I tried to expand $\exp(tAXA^{-1})$ but I am stuck, since the term inside the exponential is a mix of group and algebra matrices and I don't know how to deal with it.

 

[Orodruin]

What can you say in general for a function $f(AXA^{-1})$ assuming it can be Taylor expanded?

Edit: Or rather, what is $(A X A^{-1})^n$?

 

[atat1tata]

What can you say in general for a function $f(AXA^{-1})$ assuming it can be Taylor expanded?

Edit: Or rather, what is $(A X A^{-1})^n$?

Thanks! So:
$\exp (tAXA^{-1}) = 1 + tAXA^{-1} + \frac{t^2}{2}AX^2A^{-1} + ... = A(\exp{tX})A^{-1} \in G$

Why I was stuck, I don't understand... Thank you for the hint!